Mathematical models for drug delivery to chronic patients

Mathematically, analysis of drug delivery kinetics involves two moving boundary problems: diffusion front and eroding front. In this paper, we have models for drug delivery for the sites which can be enclosed by spherical shaped matrices covered by membranes and these problems are helpful for designing the drug delivery devices to deliver the drug inside from outside and a corresponding device supplying drug from inside. Once the time required for treatment and rate of drug delivery is known from medical diagnosis, this analysis can design a device releasing the drug/active agent over a long period of time. The purpose of such drug delivery is to achieve more effective therapies while eliminating the effect of over dosing and maintaining drug levels within the desired levels. The device may work on optimal use of drug and increase the patient’s convenience. The proposed models provide design for eroding tumor or chemotherapy to cancerous regions. The results have been obtained for steady state release rate, zero order release time and life time of the device and discussed. It has been observed that zero order time and life time increase by introducing a membrane of uniform thickness.     

Flux tracking in drug delivery

The dynamics of diffusive and stress-induced transport in polymeric delivery systems was investigated. Partial and ordinary differential equations were first written to describe drug release behaviors in Maxwell and Maxwell–Voigt materials. The time constants governing the flux and concentration responses of a permeating species were determined from a Laplace transform solution of the original model. A “tracking strategy”, based on the estimated characteristic times, was proposed to estimate the delivery rate and the concentration near the exit side of the membrane. The methodology was more efficient at times greater than the time constant and the prediction error decreased further as the process approached steady state. Numerical illustrations and comparisons made with published data show the effectiveness of the proposed approach in describing the influence of the Young modulus, viscosity and relaxation time on the transient regime.     

Mechanics of spheroidal fullerenes and carbon nanotubes for drug and gene delivery

There is considerable interest in the mechanics of carbon nanostructures,such as carbon nanotubes and fullerenes, and the manner of theirinteractions at the intermolecular level. Medical applicationsinclude the use of carbon nanotubes for targeted drug and genedelivery, for which issues relating to the acceptance and containmentof drugs or genes are not properly understood. A spheroid isan ellipsoid with two equal axes and the general spheroidalshape includes a wide variety of possible molecular configurationssuch as spheres, capped cylindrical tubes and ellipsoids ofrevolution, and therefore the determination of the interactionforces for this general shape may have many applications. Phenomenasuch as the suction of fullerenes into carbon nanotubes dueto the van der Waals interatomic interactions and ultra-lowfriction of a molecule moving inside a carbon nanotube giverise to the possibility of constructing nanoscaled oscillatorswith frequencies in the gigahertz range. This paper models themechanics of such a system by employing a six-twelve Lennard–Jonespotential taken over two surfaces assumed to be composed ofmean distributions of atoms over the two idealized surfacesof an open-ended semi-infinite circular cylinder and a spheroid.Following the methodology of previous work with spherical surfaces,the acceptance energy and suction energy for spheroidal moleculesare given and the special case of spherical molecules is alsoreproduced to validate the method. The results for ellipticalmolecules are novel and cannot be validated experimentally atthis stage, but the results for the special case of sphericalmolecules are given and shown to be in good agreement with publishedmolecular dynamical simulations. Finally, a general numerical-analyticalprocedure is proposed to calculate the Lennard–Jones potentialfor any axially symmetric surface, and the prior results obtainedfor the spheroid are used to validate the procedure.     

Mathematical modeling of drug delivery from cylindrical implantable devices

A mechanistic mathematical model applicable to the controlled dispersed‐drug release from cylindrical device such as implantable drug delivery system was derived. Analytical solutions based on the pseudosteady state approximation are derived taken account an exact external medium volume. The model prediction is accurate when the initial drug load is higher than the drug solubility in the polymer. The results obtained are compared with the analytical solutions available in the literature. The equations are corroborated by comparison with experimental profiles reported in the literature for sink conditions and non sink conditions. The evolution of concentration distribution profiles is compared for different volume of external medium. A reduction in the volume of the external solution leads to an increase in the concentration on the surface of the device, which determines decreases in the release of drug. One criterion for determining whether the volume of external solution should be considered for the prediction of drug release from cylindrical devices is established. This criterion is based on establishing a maximum percentage error allowed in the values of amount of drug released. The usefulness of the model is focused in the design of implant for controlled release of drug into a small volume of external medium of release. Copyright © 2014 John Wiley & Sons, Ltd.     

The role of microvascular tortuosity in tumor transport phenomena and future perspective for drug delivery

We tackle the fluid transport problem in vascularized tumors by solving a double Darcy model obtained via multiscale homogenization. The hydraulic conductivities of the capillary and interstitial compartments are computed solving classical problems on the representative periodic cell, which encodes details of the microvasculature. Microvascular tortuosity leads to a dramatic decrease of the capillary hydraulic conductivity, and the corresponding lowering in the pressure drop impairs tumor blood flow and consequently advection of molecules. Further perspectives for anti-cancer agents delivery are illustrated. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Algorithms for the computation of the pseudospectral radius and the numerical radius of a matrix

^** Email: mengi{at}cs.nyu.edu^*** Email: overton{at}cs.nyu.edu Two useful measures of the robust stability of the discrete-timedynamical system x_k+1 = Ax_k are the -pseudospectral radius andthe numerical radius of A. The -pseudospectral radius of A isthe largest of the moduli of the points in the -pseudospectrumof A, while the numerical radius is the largest of the moduliof the points in the field of values. We present globally convergentalgorithms for computing the -pseudospectral radius and thenumerical radius. For the former algorithm, we discuss conditionsunder which it is quadratically convergent and provide a detailedaccuracy analysis giving conditions under which the algorithmis backward stable. The algorithms are inspired by methods ofByers, Boyd–Balakrishnan, He–Watson and Burke–Lewis–Overtonfor related problems and depend on computing eigenvalues ofsymplectic pencils and Hamiltonian matrices.     

Bohr radius for the punctured disk

This paper investigates the Bohr phenomenon for the class of analytic functions from the unit disk into the punctured unit disk. The Bohr radius is shown to be 1/3.     

Numerical radius inequalities for operator matrices

We prove a numerical radius inequality for operator matrices, which improves an earlier inequality due to Hou and Du. As an application of this numerical radius inequality, we derive a new bound for the zeros of polynomials.     

Spectral radius inequalities for positive commutators

We establish several inequalities for the spectral radius of a positive commutator of positive operators in a Banach space ordered by a normal and generating cone. The main purpose of this paper is to show that in order to prove the quasi-nilpotency of the commutator we do not have to impose any compactness condition on the operators under consideration. In this way we give a partial answer to the open problem posed in the paper by J. Bra?i?, R. Drnov?ek, Y. B. Farforovskaya, E. L. Rabkin, J. Zemánek (2010). Inequalities involving an arbitrary commutator and a generalized commutator are also discussed.     

On an inequality for spectral radius

In this note, we first extend and then give a related result to an inequality involing the spectral radius of nonnegative matrices that recently appcared in the literature.     

Sharp upper and lower bounds for the Laplacian spectral radius and the spectral radius of graphs

In this paper, sharp upper bounds for the Laplacian spectral radius and the spectral radius of graphs are given, respectively. We show that some known bounds can be obtained from our bounds. For a bipartite graph G, we also present sharp lower bounds for the Laplacian spectral radius and the spectral radius, respectively.     

An algorithm for computing the numerical radius

Received on 23 October 1995. Revised on 15 July 1996. This paper is concerned with the calculation of the numericalradius of a matrix, an important quantity in the analysis ofconvergence of iterative processes. An algorithm is developedwhich enables the numerical radius to be obtained to a givenprecision, using a process which successively refines lowerand upper bounds. It uses an iteration procedure analogous tothe power method for computing the largest modulus eigenvalueof a Hermitian matrix. In contrast to that method, convergenceis possible here to a local maximum of the underlying optimizationproblem which is not global, so that only a lower bound is provided.This is used in conjunction with a technique based on the solutionof a generalized cigenvalue problem to provide an upper bound.Numerical results illustrate the performance of the method.     

On estimate for numerical radius of some contractions

For the numerical radius of an arbitrary nilpotent operator T on a Hilbert space H, Haagerup and de la Harpe proved the inequality , where n ≥ 2 is the nilpotency order of the operator T. In the present paper, we prove a Haagerup-de la Harpe-type inequality for the numerical radius of contractions from more general classes. Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 10, pp. 1335–1339, October, 2006.     

Neighborhood radius estimation for variable-neighborhood random fields

We consider random fields defined by finite-region conditional probabilities depending on a neighborhood of the region which changes with the boundary conditions. To predict the symbols within any finite region, it is necessary to inspect a random number of neighborhood symbols which might change according to the value of them. In analogy with the one-dimensional setting we call these neighborhood symbols the context associated to the region at hand. This framework is a natural extension, to d-dimensional fields, of the notion of variable length Markov chains introduced by Rissanen [24] in his classical paper. We define an algorithm to estimate the radius of the smallest ball containing the context based on a realization of the field. We prove the consistency of this estimator. Our proofs are constructive and yield explicit upper bounds for the probability of wrong estimation of the radius of the context.